Data assimilation finite element method for the linearized Navier-Stokes equations in the low Reynolds regime

In this paper, we are interested in designing and analyzing a finite element data assimilation method for laminar steady flow described by the linearized incompressible Navier-Stokes equation. We propose a weakly consistent stabilized finite element method which reconstructs the whole fluid flow from noisy velocity measurements in a subset of the computational domain. Using the stability of the continuous problem in the form of a three balls inequality, we derive quantitative local error estimates for the velocity. Numerical simulations illustrate these convergence properties and we finally apply our method to the flow reconstruction in a blood vessel.